cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 23, 69, 261, 943, 3815, 15107, 63219, 262791, 1130953, 4838813, 21185125, 92593943, 411160627, 1823656199, 8186105099, 36728532951, 166310761655
Offset: 1

Views

Author

Gus Wiseman, Jan 31 2020

Keywords

Comments

A tree-factorization of n > 1 is either (case 1) the number n itself, or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors > 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The a(1) = 1 through a(4) = 23 tree-factorizations:
  2  3      5          7
     4      6          9
     (2*2)  8          10
            (2*3)      12
            (2*4)      16
            (2*2*2)    (2*5)
            (2*(2*2))  (2*6)
                       (2*8)
                       (3*3)
                       (3*4)
                       (4*4)
                       (2*2*3)
                       (2*2*4)
                       (2*2*2*2)
                       (2*(2*3))
                       ((2*2)*4)
                       (2*(2*4))
                       (3*(2*2))
                       (4*(2*2))
                       (2*(2*2*2))
                       (2*2*(2*2))
                       ((2*2)*(2*2))
                       (2*(2*(2*2)))

Crossrefs

The orderless version is A319312.
Factorizations are A001055.
P-trees are A196545.
Twice-factorizations are A281113.
Tree-factorizations are A281118.
Enriched p-trees are A289501.
Cf. A000081 A000669, A000720, A001678, A005804, A056239, A112798, A141268, A273873, A330465.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
physemi[n_]:=Prepend[Join@@Table[Tuples[physemi/@f],{f,Select[facs[n],Length[#]>1&]}],n];
Table[Sum[Length[physemi[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,8}]
  • PARI
    \\ here TF(n) is n terms of A281118 as vector.
TF(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += w[k]^e*v[i]))); w}
a(n)={my(v=[prod(i=1, #p, prime(p[i])) | p<-partitions(n)], tf=TF(vecmax(v))); sum(i=1, #v, tf[v[i]])} \\ Andrew Howroyd, Dec 09 2020

Formula

a(n) = Sum_i A281118(A215366(n,i)).

Extensions

a(13)-a(20) from Andrew Howroyd, Dec 09 2020

A346787 Ordered lone-child-avoiding trees where vertices have decreasing subtree sizes.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 10, 19, 35, 68, 128, 253, 489, 981, 1930, 3899, 7771, 15858, 31915, 65503, 133070, 274631, 561371, 1164240, 2393652, 4983614, 10299238, 21511537, 44637483, 93552858, 194809152, 409270569, 855199845, 1800958182, 3773297872, 7963655481
Offset: 1

Views

Author

David Callan, Aug 03 2021

Keywords

Comments

a(n) is the number of size-n, rooted, ordered, lone-child-avoiding trees in which the subtrees of each non-leaf vertex, taken left to right, have weakly decreasing sizes, where size is measured by number of vertices.
The analogous trees when size is measured by number of leaves are counted by A196545.

Examples

			See Link.

Links

Crossrefs

Cf. A196545.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> b(n-1, n-2):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 05 2021
  • Mathematica
    a[1] = 1; a[2] = 0;
a[n_] /; n >= 3 := a[n] = Apply[Plus, Map[Apply[Times, Map[a, #]] &, Rest[IntegerPartitions[n - 1]]]]
Table[a[n], {n, 20}]

Formula

Counting by sizes of subtrees of the root, a(n) is the sum, over all non-singleton partitions i_1,i_2,...,i_k of n-1, of the product a(i_1)a(i_2) ... a(i_k).
G.f. satisfies A(x)=x/((1+x)*Product_{n>=1} (1 - a(n)*x^n)).

A348756 Product_{n>=1} (1 + a(n)*x^n) = 1 + x + x^2 + 2 * Sum_{n>=3} a(n)*x^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 17, 38, 84, 192, 438, 1024, 2388, 5678, 13451, 32354, 77578, 188382, 456006, 1115966, 2722174, 6702450, 16458698, 40734862, 100551658, 249995010, 619974654, 1547115954, 3852068670, 9644512918, 24095760863, 60506814284, 151622914562
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 31 2021

Keywords

Crossrefs

Cf. A032305, A196545, A348755.

A300411 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - a(k)*x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 9, 14, 19, 24, 45, 75, 105, 135, 229, 359, 503, 647, 1047, 1591, 2272, 2972, 4696, 6996, 9844, 12894, 20064, 29538, 41204, 54407, 84457, 123723, 171757, 225939, 348643, 508693, 703815, 923529, 1423892, 2076942, 2870977, 3763380, 5778379, 8414332, 11621307
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

Crossrefs

Cf. A006906, A093637, A151945, A196545, A291698, A293806, A293807, A296387.

Programs

  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - a[k] x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 42}]

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - a(n)*x^a(n)).

A348661 a(1) = 1; a(n) = Sum_{d|n, d < n} d * a(d)^(n/d).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 39, 4, 8, 1, 330, 1, 10, 9, 12495, 1, 1446, 1, 1620, 11, 14, 1, 1792050, 6, 16, 580, 10158, 1, 53002, 1, 2516534175, 15, 20, 13, 469241466, 1, 22, 17, 774558756, 1, 1696170, 1, 712914, 20160, 26, 1, 108457624531554, 8, 328588, 21, 6383964
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 28 2021

Keywords

Crossrefs

Cf. A006241, A008578 (positions of 1's), A157313, A165552, A196545, A281145.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[d < n, d a[d]^(n/d), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 52}]

Formula

For n > 1, a(n) is the coefficient of x^n/n in expansion of -log(Product_{k=1..n-1} (1 - a(k)*x^k)).
Previous Showing 81-85 of 85 results.

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